Labor Economics
1. Assume households have utility function U(C, I) = 2cl1/2, can earn wage w per hour worked, and receive V in non-wage income. Total amount of time available for leisure l and labor h is T = h + l.
(a). Set up but do not solve the household's optimization problem.
(b). Write the Lagrangian.
(c). Set up do not solve the solution condition (hint: there are two first-order conditions, plus the budget constraint).
(d). Calculate the first-order conditions.
(e). Eliminate the multiplier λ from the first order conditions. You should end up with one combined first order condition, plus the budget constraint.
(f). Isolate either C or l form the combined first-order condition. Take the expression for the isolated variable and substitute it into the budget constraint. Now solve for whichever variable you didn't isolate.
(g). Substitute back and solve for the isolated variable.
(h). Calculate the expression for the elasticity for labor supply, σ.
Suppose T = 4.000 (80 hrs non sleeping time/working week (M - F) * 50 weeks/years).
(i) Calculate the reservation wage when V = 10,000 and again when V = 50, 000. Plot the labor supply curves for each of these values of V.
(j) Calculate the supply elasticity for the following w and V: (w = 10, V = 10,000), (w = 10, V = 30,000), (w = 20, V = 10,000), (w = 20, V = 30,000).
(k) Provide intuition for the results in (i) and (j).